MASSACHUSETTS INSTITUTE of TECHNOLOGY Physics Department

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department

Physics 8.282 May 5, 2003

Problem Set 12

Due: Monday, May 12 Note: This problem set is optional. If you choose not to submit this set, it will have no eﬀect on your grade. If you turn it in to be graded, your problem set with the lowest score will be dropped. Problems 7 & 8 are marked “optional” and will not be graded even if they are turned in. Reading: Zeilik & Gregory: ﬁnish Chapter 21; read Chapters 22 and 23; start Chapter 24.

Problem 1 “Orbiting Galaxies” Zeilik & Gregory, Chapter 21, Problem #1, page 431. Refer to Figure 23–1 and Table 23–1 for relevant masses and distances. In each case, take the present separation to be equal to the semimajor axis of the orbit.

Problem 2 “Distance Determinations” Zeilik & Gregory, Chapter 22, Problem #2, page 445. Refer to Table 22–1 for a helpful summary of the basic “distance indicators”

Problem 3 “Galaxy Redshift from the Ca II K Line” Zeilik & Gregory, Chapter 22, Problem #3, page 445.

Problem 4 “Measuring a Quasar Redshift” Figure 24–15 (page 478 of Z&G) shows the spectrum of 3C 273, the ﬁrst quasistellar object (QSO) whose spectral redshift was identiﬁed (by Maarten Schmidt, 1963). If one assumes that the redshift is due to the cosmological expansion of the Universe, one can estimate the distance to the QSO by use of the empirical relationship discovered by Edwin Hubble.

a. Use the ﬁgure to determine the redshift (z = Δλ/λ0) of 3C 273. You can determine the scale of the spectrogram (in A˚ per mm) from the three wavelengths identiﬁed on the comparison spectrum. [The rest wavelength of the Hβ line is 4861 ˚A.]

1 b. Translate this redshift to a recessional velocity. c. Use the Hubble relation (with H0 = 65 km/s/Mpc) to ﬁnd the distance to 3C 273.

Problem 5 “Identifying the Lines and Determining the Redshift of a Quasar” The ﬁgure below is the visible spectrum of a distant quasar. Five prominent spectral lines can be seen in the spectrum. Also given below is a table of the laboratory wavelengths of the most observed emission lines in the spectra of quasars. Identify the lines and calculate the redshift, z = (λ − λ0)/λ0, of this quasar (λ is the observed wavelength and λ0 is the wavelength measured in the laboratory).

Hint: First measure λ for the ﬁve lines in the spectrum. Then look for ﬁve lines in the table such that λ/λ0 has nearly a constant value.

2 Problem 6 “A Simple Determination of the Hubble Law” Figure 22–1 in Z&G (page 435) shows ﬁve images of elliptical galaxies, all photographed with the same plate scale (∼ 10 arc seconds per mm). Assume, for absolute simplicity, that all of the galaxies have the same physical size (∼ 40 kpc in diameter).

a. Measure the diameter of each galaxy and determine its distance in Mpc. b. Compare your answers to the values given between the two columns of ﬁgures. c. Plot the observed Doppler velocities (given in the righthand column of spectra) vs. the distances to these galaxies as determined in part (a). d. Determine the Hubble constant, H0, from your graph.

Problem 7 Optional “FreeFall Time for a Cluster of Galaxies” Zeilik & Gregory, Chapter 23, Problem #5, page 463. Take the following as a model for the cluster: Number of galaxies in cluster = 1000 11 Mass of each galaxy ∼ 10 M � Radius of the cluster of galaxies ∼ 6 Mpc Procedure for making a crude estimate of the freefall timescale: a. Write the equation of motion for a galaxy near the edge of the cluster; (assume that the 1000 galaxies are distributed in space so that they eﬀectively form a spherically symmetric mass distribution) b. Use dimensional arguments to ﬁnd a characteristic time for the collapse.

Problem 8 Optional “Luminosity Function of Galaxies in Rich Clusters” Zeilik & Gregory, Chapter 23, Problem #6, page 463. Note that the units of φ are galaxies per Mpc3, and that φ(L) = φ∗ × (L/L∗)−5/4 .